I wonder if you can help me out with the following problem:
Find all $u \in \mathcal{S}(\mathbb{R^n})$ such that there is an $f \in \mathcal{S}(\mathbb{R^n})$ for which $u- u \ast f= f \ast f$.
I think it should be solved using the fourier transform but I'm not sure how. Since $u,f \in \mathcal{S}(\mathbb{R^n})$ we have $u \ast f, f \ast f \in \mathcal{S}(\mathbb{R^n})$ and one is allowed to take the Fourier transform to get $\hat{f}(\xi)^2, \hat{u}(\xi)\hat{f}(\xi) \in \mathcal{S}(\mathbb{R^n})$ which gives the equation $\hat{u}(1-\hat{f})=\hat{f}^2$. I'm not really sure what to do with this though (or if it's the right way to go). Do you have any ideas?